Empirical and Instance-Dependent Estimation of Markov Chain and Mixing Time

We tackle the problem of estimating the mixing time of a Markov chain from a single trajectory of observations. In contrast with previous works which considered Hilbert space methods to estimate spectral gaps, we opt for an approach based on contraction with respect to total variation. Specifically, we define and estimate a generalized contraction coefficient based on Dobrushin's. We show that this quantity -- unlike the spectral gap -- controls the mixing time up to strong universal constants and remains valid for non-reversible chains. We design fully data-dependent confidence intervals around the coefficient, which are both easier to compute and thinner than their spectral counterparts. Furthermore, we initiate the beyond worst-case analysis, by showing how to leverage additional information about the transition matrix in order to obtain instance-dependent rates for its estimation with respect to the induced uniform norm, as well as some of its mixing properties.